which implies the first and third characteristic fields are genuinely nonlinear and the second field is linearly degenerate.
The basic theory of hyperbolic systems of conservation laws (for example, see Ref. [1]}) implies that for any given constants \( (v_-, u_-, \theta _-) \)  with \( v_->0 \) , \( \theta _->0 \) , there exists a suitable neighborhood \( \Omega (v_-, u_-, \theta _-) \)  of \( (v_-, u_-, \theta _-) \)  such that for any \( (v_+, u_+, \theta _+) \in \Omega (v_-, u_-, \theta _-) \) ,
the Riemann problem \( (\ref {nstuidao2}) \)  and \( (\ref {Riemanchuzhi}) \)  admits a solution consisting of the basic wave patterns.
